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Quasinormal modes of a massive Dirac field were calculated for various static black hole backgrounds with the help of the WKB formula. These estimations, however, are rough and valid only for very small values of mu M, where M and mu are mass of the black hole and field respectively. Thus, no accurate calculations of massive Dirac modes are known even for the Schwarzschild black hole and this is all the more so for the Kerr solution. Here we calculate quasinormal modes of a massive Dirac field in the Kerr background. We have shown that the infinitely long-lived quasinormal modes (quasiresonances), which exist for boson fields, appear also in the fermions' quasinormal spectrum. Two chiralities of massive fermions lead to an additional 'fine structure' in the spectrum. We discuss the effect of this fine structure on the behavior of quasiresonances and the stability. The analysis is also extended to a charged massive field in the Kerr-Newman background.